In practice, it is unfortunately difficult to entirely factorize or diagonalize large sparse matrices using dense solvers based on Gaussian elimination, Givens rotations, or Householder reflections due to memory limitations. Krylov subspace projection (KSP) methods are commonly used to obtain solutions for sparse high-dimensional linear systems. The approximations to these solutions are estimated by minimizing the residual over the subspace formed. A well-known KSP method is the conjugate gradient (CG) [1]}, which is used for solving linear systems involving symmetric and positive definite matrices. For symmetric and possibly indefinite system, iterative method like minimum residual (MINRES) method is rather used [2]}. In case of non-symmetric matrices, the biconjugate gradient stabilized (BiCGSTAB) method [3]}, [4]} that is a generalized CG method, and also a generalized minimum residual (GMRES) method [4]} are available. KSP methods can also be employed to extract a few pairs of eigenvectors and eigenvalues iteratively, for instance, the Krylov-Schur method [6]}, [7]}.
